void fix_peripheral_orientations(
Triangulation *manifold)
{
Tetrahedron *tet;
VertexIndex v;
FaceIndex f;
Cusp *cusp;
/*
* This function should get called only for orientable manifolds.
*/
if (manifold->orientability != oriented_manifold &&
manifold->orientability != oriented_orbifold)
uFatalError("fix_peripheral_orientations", "orient");
/*
* Compute the intersection number of the meridian and longitude.
*/
copy_curves_to_scratch(manifold, 0, FALSE);
copy_curves_to_scratch(manifold, 1, FALSE);
compute_intersection_numbers(manifold);
/*
* Reverse the meridian on cusps with intersection_number[L][M] == -1.
*/
/* which Tetrahedron */
for (tet = manifold->tet_list_begin.next;
tet != &manifold->tet_list_end;
tet = tet->next)
/* which ideal vertex */
for (v = 0; v < 4; v++)
if (tet->cusp[v]->intersection_number[L][M] == -1)
/* which side of the vertex */
for (f = 0; f < 4; f++)
if (v != f)
{
tet->curve[M][right_handed][v][f] = - tet->curve[M][right_handed][v][f];
if (tet->curve[M][left_handed][v][f] != 0.0
|| tet->curve[L][left_handed][v][f] != 0.0)
uFatalError("fix_peripheral_orientations", "orient");
}
/*
* When we reverse the meridian we must also negate the meridional
* Dehn filling coefficient in order to maintain the same (oriented)
* Dehn filling curve as before. However, this Dehn filling curve
* will wind clockwise around the core geodesics, relative to
* the global orientation on the manifold (because the global
* orientation disagrees with the local orientation we had been using
* on the nonorientable manifold's torus cusp). This forces a whole
* new solution to be found to the gluing equations. To avoid this,
* we reverse the direction of the Dehn filling curve (i.e. we
* negate both the m and l coefficients). The net effect is that
* we negate the l coefficient.
*
* This reversal of the Dehn filling curve is not really
* necessary, and could be eliminated if it's ever causes problems.
*/
for (cusp = manifold->cusp_list_begin.next;
cusp != &manifold->cusp_list_end;
cusp = cusp->next)
if (cusp->intersection_number[L][M] == -1)
cusp->l = - cusp->l;
}
static void compute_cusp_map(
Triangulation *manifold0,
Triangulation *manifold1,
Isometry *isometry)
{
Tetrahedron *tet;
VertexIndex v;
int i;
/*
* Copy the manifold1's peripheral curves into
* scratch_curves[0], and copy the images of manifold0's
* peripheral curves into scratch_curves[1].
*
* When the manifold is orientable and the Isometry is
* orientation-reversing, and sometimes when the manifold
* is nonorientable, the images of the peripheral curves
* of a torus will lie on the "wrong" sheet of the Cusp's
* orientation double cover. (See peripheral_curves.c for
* background material.) Therefore we copy the images of
* the peripheral curves of torus cusps to both sheets of
* the orientation double cover, to guarantee that the
* intersection numbers come out right.
*/
copy_curves_to_scratch(manifold1, 0, FALSE);
copy_images_to_scratch(manifold0, 1, TRUE);
/*
* Compute the intersection numbers of the images of manifold0's
* peripheral curves with manifold1's peripheral curves..
*/
compute_intersection_numbers(manifold1);
/*
* Now extract the cusp_maps from the linking numbers.
*
* There's a lot of redundancy in this loop, but a trivial
* computation so the redundancy hardly matters.
*
* Ignore negatively indexed Cusps -- they're finite vertices.
*/
for (tet = manifold0->tet_list_begin.next;
tet != &manifold0->tet_list_end;
tet = tet->next)
for (v = 0; v < 4; v++)
if (tet->cusp[v]->index >= 0)
for (i = 0; i < 2; i++) /* i = M, L */
{
isometry->cusp_map[tet->cusp[v]->index][M][i]
= + tet->image->cusp[EVALUATE(tet->map, v)]->intersection_number[L][i];
isometry->cusp_map[tet->cusp[v]->index][L][i]
= - tet->image->cusp[EVALUATE(tet->map, v)]->intersection_number[M][i];
}
}
static void adjust_Klein_cusp_orientations(
Triangulation *manifold)
{
/*
* As explained at the top of this file, a Cusp's peripheral curves
* live in the its orientation double cover. When I first wrote this
* file, I didn't worry about the orientation of peripheral curves
* in nonorientable manifolds. Subsequently it became clear that
* they should have the standard orientation relative to the Cusp's
* (oriented, not just orientable) orientation double cover. In
* the case of a torus Cusp, the peripheral curves live in one
* (arbitrarily chosen) component of the orientation double cover;
* in the case of a Klein bottle Cusp, the orientation double cover
* is connected.
*
* Fortunately, it's very easy to check whether the peripheral curves
* have the standard orientation, and to correct them if necessary.
* The definition of the standard orientation for peripheral curves on
* a torus is that when the fingers of your right hand point in the
* direction of the meridian and your thumb points in the direction
* of the longitude, the palm of your hand should face the cusp and
* the back of your hand should face the fat part of the manifold.
* Combining this with the definition of the intersection number
* found at the top of intersection_numbers.c reveals that the
* intersection number of the longitude and the meridian (in that
* order) should be +1. If it happens to be -1, we must reverse
* the meridian.
*/
/*
* If the manifold is oriented, then the peripheral curves will
* already have the correct orientation. In fact, they will lie
* on the right handed sheet of the Cusp's orientation double
* cover, relative to the orientation of the manifold.
*/
if (manifold->orientability == oriented_manifold)
return;
/*
* The scratch curves might already be in use, so let's make
* a copy of whatever's there.
*/
backup_scratch_curves(manifold);
/*
* Copy the peripheral curves to both sets of scratch_curve fields.
*/
copy_curves_to_scratch(manifold, 0, FALSE);
copy_curves_to_scratch(manifold, 1, FALSE);
/*
* Compute their intersection numbers.
*/
compute_intersection_numbers(manifold);
/*
* Restore whatever used to be in the scratch_curves.
*/
restore_scratch_curves(manifold);
/*
* On Cusps where the intersection number of the longitude and
* meridian is -1, reverse the meridian.
*/
reverse_meridians_where_necessary(manifold);
}
